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Applications of approximate gradient schemes for nonlinear parabolic equations

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Abstract

We develop gradient schemes for the approximation of the Perona-Malik equations and nonlinear tensor-diffusion equations. We prove the convergence of these methods to the weak solutions of the corresponding nonlinear PDEs. A particular gradient scheme on rectangular meshes is then studied numerically with respect to experimental order of convergence which shows its second order accuracy. We present also numerical experiments related to image filtering by time-delayed Perona-Malik and tensor diffusion equations.

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Authors and Affiliations

  1. Université Paris-Est, 5 boulevard Descartes, Cité Descartes, Champs-sur-Marne, 774 54, Marne-la-Vallée, Cedex 2, France

    Robert Eymard

  2. Department of Mathematics and Constructive Geometry, Slovak University of Technology, Radlinského 11, Bratislava, Slovakia

    Angela Handlovičová

  3. Centre de Mathématiques et Informatique (CMI), Aix-Marseille Université, Technopôle Château-Gombert, 39, rue F. Joliot Curie, 134 53, Marseille Cedex 13, France

    Raphaèle Herbin

  4. Department of Mathematics and Constructive Geometry, Slovak University of Technology, Radlinského 11, Bratislava, Slovakia

    Karol Mikula & Olga Stašová

Authors
  1. Robert Eymard
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  2. Angela Handlovičová
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  3. Raphaèle Herbin
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  4. Karol Mikula
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  5. Olga Stašová
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Corresponding author

Correspondence to Robert Eymard.

Additional information

The research of A. Handlovičová, K. Mikula and O. Stašová has been supported by grants APVV-0184-10, VEGA 1/1063/11 and VEGA 1/1137/12.

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Eymard, R., Handlovičová, A., Herbin, R. et al. Applications of approximate gradient schemes for nonlinear parabolic equations. Appl Math 60, 135–156 (2015). https://doi.org/10.1007/s10492-015-0088-4

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  • DOI: https://doi.org/10.1007/s10492-015-0088-4

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